There's an imaginary $i$ in the Schrödinger equation, which I guess is to define the position of the particle in a space-time involving a complex function. But what is the real physical significance of $i$ in the equation?
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4It makes it a wave equation. Take the imaginary factor out and the equation changes to what looks like the heat equation with an additional potential. This fundamentally changes the quality of the solution. The Schroedinger equation does not describe the position of a particle. It describes the wave function of a single particle. – CuriousOne Mar 14 '16 at 06:57
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1i guess wave function signifies the position/state of particle in space-time.....si=Ae^i(kx-wt)....in general form.... – Mar 14 '16 at 06:59
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If you look at it in terms of a wave description, then you can find an analogy with the Fourier transform from time domain to frequency domain.There, an i is also included in order to make the mathematical formalism work consistently. – Santiago Mar 14 '16 at 07:42
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1@AmritSharma: You'll get that if there were not $i$ probability would leak exponentially; but it is $i$ which actually gives probability amplitude leakage the wave nature. Check this related question I asked a few times back; it shows the significance of $i$, though implicitly: Understanding “Propagation in a Crystal Lattice”: What is the difference between 'amplitude leakage' & 'probability leakage'?. – Mar 14 '16 at 09:24
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@CuriousOne: For God' sake, stop writing answers in the comment; you always say right and answer to the point but damn they are mostly comments : ( – Mar 14 '16 at 09:26
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5related (or duplicate?): Where does the $i$ come from in the Schrödinger equation? – AccidentalFourierTransform Mar 14 '16 at 10:06
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The Schrödinger equation is related to the Schrödinger-Pauli equation (which has spin).
And there are relativistic equations too. And they all have algebraic objects that add and multiply like $\pm i$ do. So you can look at those equations and their factors and terms and see what they represent and look at the nonrelativistic and constant spin limits.
One geometric description of the relativistic wave equation for an object of spin 1/2 is that it takes a reference spatial plane (which adds and multiplies like $\pm i$) and gives it a phase rotation in its plane, scales it by a positive scalar, rotates it into an arbitrary attitude plane (possibly different than the original reference plane's attitude), and gives it a relativistic boost to put the (now rotated) reference spin plane into an arbitrary plane of simultaneity. That covers all the degrees of freedom of the equation and does it will things that algebraically act like those parts of the equation and geometrically do those things to geometric objects in a spacetime.
Then the non relativistic limit has a constant spin limit that is the Schrödinger equation where the $i$ survives in the equation as the initial arbitrary reference plane.
That doesn't mean that's what the $i$ is. But for some problems you are using the Schrödinger equation for a spin 1/2 particle in a a nonrelativistic situation with a constant spin. In which case it is harder to think it can represent anything else.
Timaeus
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